Math Is Fun Forum

mattlim

Guest

Geometry Help!(Specifically Analytical Geo)

Help! I've been trying these problems for some time, but I haven't been able to find a working method to solve them.

1. Find the largest real number x for which there exists a real number y such that x^2 + y^2 = 2x + 2y.

2. The lines y = (5/12)x and y = (4/3)x are drawn in the coordinate plane. Find the slope of the line that bisects the angle between these lines.

3. Let A = (1,2), B = (0,1), and C = (5,0). There exists a point Q and a constant k such that for any point P, PA^2 + PB^2 + PC^2 = 3PQ^2 + k. Find the constant k.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Geometry Help!(Specifically Analytical Geo)

Hi;

1)

I am getting 

2)

You can now get the slope.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

Re: Geometry Help!(Specifically Analytical Geo)

hi mattlim

Welcome to the forum.

Q1. Because the x^2 and y^2 terms have equal positive coefficients, that is the equation of a circle.  So work it round to the standard form for a circle.

http://www.mathsisfun.com/algebra/circle-equations.html

add 1 + 1 to each side.

So the centre is at (1,1) and the radius is root 2.  So you need to work out the coordinate of the 'rightmost' point on the circumference.

Q2.  If you call the angles of the lines to the x axis A and B

tan(A) = 4/3 and tan(B) = 5/12

The angle of the bisector will be (A+B)/2. 

You can work this out using the following tan formulas

and

http://www.mathsisfun.com/algebra/trigo … ities.html

Q3.

Let A = (1,2), B = (0,1), and C = (5,0). There exists a point Q and a constant k such that for any point P, PA^2 + PB^2 + PC^2 = 3PQ^2 + k.

So call P the point (x,y) and Q the point (h,j) and form the equation:

Simplify carefully and you should be able to find h, j, and k.

(Because this must be true for all x and all y you can set equal the x terms to get one equation; the y terms to get a second and the non x,y terms to get a third. You will find all the squared terms all cancel out so you can set the x terms equal to get h, the y terms to get j and finally the non-x,y terms to get k.)

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob